Random surface interpretations of two-dimensional Liouville quantum gravity and Yang-Mills theory
Name
park-minj-phd-math-2022-thesis.pdf
Description
Thesis PDF
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3.6 MB
Format
Adobe PDF
Checksum (MD5)
7514ec44a9ec67cf9c0365b87fc5c3a7
Author(s)
Park, Minjae
Advisor(s)
Sheffield, Scott R.
Date Issued
May 2022
Publisher
Massachusetts Institute of Technology
Abstract
The theory of random surfaces (or "sums over surfaces") has its historical roots in quantum gravity, string theory, statistical physics, and combinatorics. This thesis explores random surfaces in two settings: one related to Liouville quantum gravity, and one related to Euclidean Yang-Mills theory in two dimensions.
The first part introduces a specific regularization of Liouville quantum gravity surfaces. It also establishes the Polyakov-Alvarez formula on non-smooth surfaces with Brownian loops instead of the zeta-regularized Laplacian determinant. Consequently, "weighting by a Brownian loop soup" changes the so-called central charge of the regularized random surfaces, as expected in physic literature. This result justifies a definition of Liouville quantum gravity surfaces in the supercritical regime where the central charge is greater than 1.
The second part describes continuum Wilson loop expectations on the plane as sums over surfaces, an example of gauge string duality. In contrast to the Gross-Taylor expansion, our weight is explicit as ±Nᵡ where χ is the Euler characteristic, for any gauge group U(N), SO(N), Sp(N/2). Based on the well-established continuum theory in two dimensions, we provide a probabilistic treatment for Wilson loop expectations, also leading to various applications like an alternative proof for the Makeenko-Migdal equation and a connection with a random walk on permutations.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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